86 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA For D(ft(x)) is the limit, as h ^ 0 tends to 0, of the mapping and the result therefore follows from (12.16.5). The mapping f'to is said to be the derivative at the point t0 of the mapping tt-+ft (with respect to the topology of simple convergence), or the weak derivative when F is the field of scalars (and therefore &(E; F) = E'). Remark (12.16.6.1) Let A be an open set in C and let zi-»/z be a weakly analytic mapping (12.15) of A into the dual E' of a complex Frechet space F. Then the same argument as in (12.16.6) shows that there exists a weak derivative zt-+fz of z\-*fz9 with fz(x) = D(fz(x)) for all xe E, and this weak derivative is weakly analytic. Moreover, for each a e A and each circuit y contained in A — {a}, we have the Cauchy formula (12.16.6.2) ;(* ;?)/«(*) = J^ {fz(x)dz 2nijy z-a Conversely, let y be a road in C defined in an interval I = \b, c] in R, and let z\-+gz be a weakly continuous mapping of y(I) into E'. Then, for z the mapping (12.16.6.3) f 9- Jy C is a linear form/z on E, belonging to E'. Indeed, for each x e E the right-hand side of (12.16.6.3) is the limit of a sequence n jt=o U - z where and ((3.16.5) and (8.7.8)), hence our assertion is a consequence of the Banach- Steinhaus theorem. By virtue of (9.9.2), fz is weakly analytic in C — y(I), and for a $ y(I) we can write (12.16.6.4) fjx) = f CB(X)(Z - a)", n = 0 the series being convergent in any disk with center a not meeting y(I). The of extension of identities.